On the distance of finite numbers of given length (Q1094447)

Let \(1<q<2\), \(L:=1/(q-1)\). Every \(x\in [0,L]\) has the unique representation \(x=\sum_{j=1}^{\infty} \varepsilon_j(x)/q^j\) with \(\varepsilon_j(x)\in \{0,1\}\), \(j=1,2,\dots\) and \(\varepsilon_n\) the greatest of the two values 0 and 1 such that \(\sum_{j=1}^n \varepsilon_j(x)/q^j\leq x\), \(n=1,2,\dots\). If \(\varepsilon_N(x)=1\) and \(\varepsilon_n(x)=0\) for \(n>N\), then \(x\) is called a finite number of length \(N\). Denote by \(U_N\) the set of all finite numbers of length \(N\) and define \(\Delta_N(x):=\min_{y\in U_N}| x-y|\), \(\delta_N:=\sup_{x\in [0,L]}\delta_N(x)\). The main result of the paper says that for every \(N\geq 1\) one has \[ \delta_N\leq \frac{3L(L+1)\log_2 2N}{q^N}. \] The question whether this estimate can be improved is also discussed. It is shown e.g. that there exists a number \(q\) for which \(\limsup_{N\to \infty}(\Delta_N q^N/\log N)>0\). The proofs are rather specialistic but very clearly presented.